Let Sym ⁡ M \operatorname {Sym} M be the symmetric group of an infinite set M M . What is the smallest subgroup of Sym ⁡ M \operatorname {Sym} M containing a given element if the subgroup is subject to the further condition that it is also the automorphism group of some finitary algebra on M M ? The structures of such closed hulls are related to the disjoint-cycle decompositions of the given elements. If the closed hull is not just the cyclic subgroup on the given element then it is nonminimal as a closed hull and is represented as a subdirect product of finite cyclic groups as well as by a quotient group of a group of infinite sequences. We determine the conditions under which it has a nontrivial primary component for a given prime p p and show that such components must be bounded abelian groups.